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web.groovymark@gmail.com
- November 15, 2024
Question 41
What is De Morgan’s Law in logic?
a) A set of rules that describe how negations interact with conjunctions and disjunctions
b) A set of rules that describe how conjunctions interact with disjunctions
c) A set of rules that describe how propositions are logically equivalent
d) A set of rules that describe how tautologies are formed
Correct Answer: a) A set of rules that describe how negations interact with conjunctions and disjunctions
Explanation: De Morgan's Law describes how negations distribute over conjunctions and disjunctions, specifically: ¬(p∧q) = ¬p∨¬q and ¬(p∨q) = ¬p∧¬q.
Question 42
Which of the following is true for an existential quantifier?
a) It asserts that there exists at least one element in the domain for which the predicate is true
b) It asserts that the predicate is true for all elements in the domain
c) It asserts that the predicate is false for all elements in the domain
d) It asserts that the predicate is true for no elements in the domain
Correct Answer: a) It asserts that there exists at least one element in the domain for which the predicate is true
Explanation: The existential quantifier ∃x asserts that there is at least one element in the domain for which the predicate P(x) is true.
Question 43
What does the universal quantifier assert?
a) The predicate is true for all elements in the domain
b) The predicate is true for some elements in the domain
c) The predicate is false for all elements in the domain
d) The predicate is false for some elements in the domain
Correct Answer: a) The predicate is true for all elements in the domain
Explanation: The universal quantifier ∀x asserts that the predicate P(x) is true for every element in the domain.
Question 44
Which of the following describes a proof by contradiction?
a) Assuming the negation of the statement to be proved and deriving a contradiction
b) Assuming the statement to be proved and directly deriving the conclusion
c) Proving a statement by dividing the proof into multiple cases
d) Proving a statement by showing that no counterexample exists
Correct Answer: a) Assuming the negation of the statement to be proved and deriving a contradiction
Explanation: A proof by contradiction assumes the negation of the statement to be proved and then shows that this assumption leads to a contradiction, thereby proving the original statement.
Question 45
Which of the following describes a direct proof?
a) Proving a statement by directly showing that the hypothesis leads to the conclusion
b) Proving a statement by assuming the negation and deriving a contradiction
c) Proving a statement by dividing the proof into multiple cases
d) Proving a statement by showing that no counterexample exists
Correct Answer: a) Proving a statement by directly showing that the hypothesis leads to the conclusion
Explanation: A direct proof involves starting with the hypothesis and making logical steps that lead to the conclusion.
Question 46
What is the purpose of a proof by contrapositive?
a) To prove a statement by proving its contrapositive, ¬q→¬p
b) To prove a statement by assuming its negation and deriving a contradiction
c) To prove a statement by showing it holds for all cases
d) To prove a statement by finding a counterexample
Correct Answer: a) To prove a statement by proving its contrapositive, ¬q→¬p
Explanation: A proof by contrapositive involves proving the contrapositive of a statement, which is logically equivalent to proving the original statement.
Question 47
Which of the following best describes a proof by exhaustion?
a) A proof that verifies a statement by checking all possible cases
b) A proof that assumes the negation of the statement and derives a contradiction
c) A proof that uses the contrapositive to establish the truth of a statement
d) A proof that divides the domain into multiple cases
Correct Answer: a) A proof that verifies a statement by checking all possible cases
Explanation: A proof by exhaustion involves checking all possible cases to verify that the statement holds for each one.
Question 48
Which of the following describes a mathematical induction proof?
a) A proof that involves a base case and an inductive step
b) A proof that assumes the negation of the statement and derives a contradiction
c) A proof that verifies a statement by checking all possible cases
d) A proof that uses a counterexample to disprove a statement
Correct Answer: a) A proof that involves a base case and an inductive step
Explanation: A proof by mathematical induction involves proving a base case and then showing that if the statement holds for one case, it holds for the next case.
Question 49
Which of the following is the base case in mathematical induction?
a) The initial case for which the statement is proven to be true
b) The final case for which the statement is proven to be true
c) The case where the statement is proven to be false
d) The case where the statement is assumed to be true
Correct Answer: a) The initial case for which the statement is proven to be true
Explanation: The base case in mathematical induction is the initial case where the statement is directly proven to be true, forming the foundation for the inductive step.
Question 50
Which of the following is true for strong mathematical induction?
a) The inductive step assumes the statement is true for all previous cases up to the current one
b) The inductive step assumes the statement is true for only one previous case
c) The base case is omitted in strong induction
d) The statement is assumed to be false and then proven true
Correct Answer: a) The inductive step assumes the statement is true for all previous cases up to the current one
Explanation: In strong mathematical induction, the inductive step assumes that the statement is true for all previous cases up to the current one, rather than just for one prior case.
